Mathematical Sciences: Semiliner Partial Different EquationsCurve Shortening in Minkowski Geometry and Branching Processes in Probability

数学科学：半线性偏微分方程闵可夫斯基几何中的曲线缩短和概率中的分支过程

• 批准号: 9225145
• 负责人: Yi Li
• 金额: $ 4万
• 依托单位: University of Rochester
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1995-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9225145&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Semiliner; Partial; Different

Li 9225145 This project concentrates on three areas of research in mathematical analysis. The general theme is that of determining existence, regularity, local and global behavior and symmetry properties of solutions of nonlinear equations and systems. In particular, work on semilinear elliptic partial differential equations will continue, examining the Lane-Emden equation and Matukuma equation of astrophysics. This includes a complete classification of positive solutions, and to determine whether all solutions are radial with limiting values of zero or infinity. A second line of investigation considers anisotropic curve shortening. This has a more geometric flavor in which embedded plane curves evolve according to a rate proportional to the normal vector. The present work focuses on a generalized setting of curves on manifolds in a Minkowski geometry. Efforts will be made to determine when such flows can shrink to a point when standing symmetry hypotheses are not present. The ideas represented by this work can be viewed as models for the evolution of phase boundaries such as the boundaries of growing crystals. The third direction this works takes involves measure-valued branching processes, including the model on the evolution caused by point catalysts. *** The work has its roots in stochastic partial differential equations arising in the study of spatially distributed birth and death processes. A particular goal is to analyze nonequilibrium reactions in the presence of multiple catalysts.

Li 9225145 该项目集中于数学分析的三个研究领域。 总的主题是确定非线性方程和系统解的存在性、规律性、局部和全局行为以及对称性。特别是，半线性椭圆偏微分方程的研究将继续进行，研究天体物理学的 Lane-Emden 方程和 Matukuma 方程。 这包括正解的完整分类，并确定所有解是否都是极限值为零或无穷大的径向解。第二项研究考虑各向异性曲线缩短。 这具有更多的几何风味，其中嵌入的平面曲线根据与法向量成比例的速率演变。目前的工作重点是闵可夫斯基几何中流形上曲线的广义设置。 将努力确定当不存在常设对称性假设时，此类流量何时可以收缩到某个点。 这项工作所代表的想法可以被视为相边界演化的模型，例如生长晶体的边界。 这项工作的第三个方向涉及测值分支过程，包括点催化剂引起的演化模型。 *** 这项工作源于研究空间分布的出生和死亡过程中产生的随机偏微分方程。 一个特定的目标是分析多种催化剂存在下的非平衡反应。